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An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

George S, Pareth S

Journal of Applied Analysis

2013

Vol. 19, nr 2

181--196

Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero X* of a differentiable function F defined on a convex set D of a Hilbert space H, we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation F(x)=f, where F : D(F) ⊆ X → X is a nonlinear monotone operator defined on a real Hilbert space X. It is assumed that F(∧x)=f and that the only available data are fδ with ||f- fδ||≤δ. We prove that the TSNLM converges cubically to a solution of the equation F(x)+α(x-x0)= fδ (such solution is an approximation of ∧x) where x0 is the initial guess. Under a general source condition on x0-∧x, we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

A common fixed points existence theorem for monotone operators in locally convex space with a cone

Mosjagin V., Sirokov B.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

2000

z. 24 [181]

55-58

It is proved that there exist common fixed points for monotone operators in locally convex spaces with a cone.